System Of Linear Equations
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For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
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System Of Linear Equations
- 2. 1.1 Introduction to Systems of Linear Equations a linear equation in n variables: 332211 bxaxaxaxa nn =++++ a1 ,a2 ,a3 ,…,an , b: real number a1 : leading coefficient x1 : leading variable Notes: (1) Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential, or logarithmic functions. (2) Variables appear only to the first power.
- 3. Notes: Every system of linear equations has either (1) exactly one solution, (2) infinitely many solutions, or (3) no solution.
- 4. Equivalent: Two systems of linear equations are called equivalent if they have precisely the same solution set. Notes: Each of the following operations on a system of linear equations produces an equivalent system. (1) Interchange two equations. (2) Multiply an equation by a nonzero constant. (3) Add a multiple of an equation to another equation.
- 5. (4) For a square matrix, the entries a11, a22, …, ann are called the main diagonal entries. 1.2 Gaussian Elimination and Gauss-Jordan Elimination m×n matrix: mnmmm n n n aaaa aaaa aaaa aaaa 321 3333231 2232221 1131211 rowsm columnsn (3) If , then the matrix is called square of order n.nm = Notes: (1) Every entry aij in a matrix is a number. (2) A matrix with m rows and n columns is said to be of size m×n .
- 6. Ex 1: Matrix Size ]2[ 00 00 − 2 1 031 − 47 22 πe 11× 22× 41× 23× Note: One very common use of matrices is to represent a system of linear equations.
- 7. a system of m equations in n variables: mnmnmmm nn nn nn bxaxaxaxa bxaxaxaxa bxaxaxaxa bxaxaxaxa =++++ =++++ =++++ =++++ 332211 33333232131 22323222121 11313212111 = mnmmm n n n aaaa aaaa aaaa aaaa A 321 3333231 2232221 1131211 = mb b b b 2 1 = nx x x x 2 1 bAx =Matrix form:
- 9. Elementary row operation: jiij RRr ↔:(1) Interchange two rows. ii k i RRkr →)(:)( (2) Multiply a row by a nonzero constant. jji k ij RRRkr →+)(:)( (3) Add a multiple of a row to another row. Row equivalent: Two matrices are said to be row equivalent if one can be obtained from the other by a finite sequence of elementary row operation.
- 10. Row-echelon form: (1, 2, 3) (1) All row consisting entirely of zeros occur at the bottom of the matrix. (2) For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). (3) For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. Reduced row-echelon form: (1, 2, 3, 4) (4) Every column that has a leading 1 has zeros in every position above and below its leading 1.
- 11. form)echelon -row(reduced form) echelon-(row Ex 4: (Row-echelon form or reduced row-echelon form) − − 2100 3010 4121 − −− 10000 41000 23100 31251 0000 3100 5010 − 0000 3100 2010 1001 − − − 3100 1120 4321 − − 4210 0000 2121 form) echelon-(row form)echelon -row(reduced
- 12. Gaussian elimination: The procedure for reducing a matrix to a row-echelon form. Gauss-Jordan elimination: The procedure for reducing a matrix to a reduced row-echelon form. Notes: (1) Every matrix has an unique reduced row echelon form.(2) A row-echelon form of a given matrix is not unique. (Different sequences of row operations can produce different row-echelon forms.)
- 13. −− − − 456542 1280200 28124682 12r The first nonzero column Produce leading 1 Zeros elements below leading 1 leading 1 Produce leading 1 The first nonzero column Ex: (Procedure of Gaussian elimination and Gauss-Jordan elimination) −− − − 456542 28124682 1280200 −− − − 456542 1280200 1462341)( 1 2 1 r −− − − 24170500 1280200 1462341)2( 13 − r Submatrix
- 14. Zeros elements below leading 1 Zeros elsewhere leading 1 Produce leading 1 leading 1 form)echelon-(row −− −− − 24170500 640100 1462341)( 2 2 1− r −− − 630000 640100 1462341)5( 23 − r −− − 210000 640100 1462341 ) 3 1 ( 3r Submatrix form)echelon-(row form)echelon-row(reduced − 210000 200100 202341)4( 32r 210000 200100 802041)3( 21r −− − 210000 640100 202341)6( 31 − r form)echelon-(row
- 15. Ex 7: Solve a system by Gauss-Jordan elimination method (only one solution) 17552 43 932 =+− −=+− =+− zyx yx zyx Sol: matrixaugmented − −− − 17552 4031 9321 −−− − 1110 5310 9321)2( 13 )1( 12 , − rr − 4200 5310 9321)1( 23r − 2100 1010 1001)9( 31 )3( 32 )2( 21 ,, −− rrr 2 1 1 = −= = z y x form)echelon-(row form)echelon-row(reduced − 2100 5310 9321 ) 2 1 ( 3r
- 16. Ex 8 ： Solve a system by Gauss-Jordan elimination method (infinitely many solutions) 153 0242 21 311 =+ =−+ xx xxx −− 1310 2501 )2( 21 )1( 2 )3( 12 )( 1 ,,,2 1 −−− rrrr isequationsofsystemingcorrespondthe 13 25 32 31 −=− =+ xx xx 3 21 variablefree ,variableleading x xx ： ： Sol: − 1053 0242 matrixaugmented form)echelon -row(reduced
- 17. 32 31 31 52 xx xx +−= −= Let tx =3 , ,31 ,52 3 2 1 tx Rttx tx = ∈+−= −= So this system has infinitely many solutions.
- 18. Homogeneous systems of linear equations: A system of linear equations is said to be homogeneous if all the constant terms are zero. 0 0 0 0 332211 3333232131 2323222121 1313212111 =++++ =++++ =++++ =++++ nmnmmm nn nn nn xaxaxaxa xaxaxaxa xaxaxaxa xaxaxaxa
- 19. Trivial solution: Nontrivial solution: other solutions 0321 ===== nxxxx Notes: (1) Every homogeneous system of linear equations is consistent. (2) If the homogenous system has fewer equations than variables, then it must have an infinite number of solutions.(3) For a homogeneous system, exactly one of the following is true. (a) The system has only the trivial solution. (b) The system has infinitely many nontrivial solutions in addition to the trivial solution.
- 20. Ex 9: Solve the following homogeneous system 032 03 321 321 =++ =+− xxx xxx − 0110 0201 )1( 21 )( 2 )2( 12 ,, 3 1 rrr − Let tx =3 Rttxtxtx ∈==−= ,,,2 321 solution)(trivial0,0When 321 ==== xxxt Sol: − 0312 0311 matrixaugmented form)echelon -row(reduced 3 21 variablefree ,variableleading x xx ： ：
- 21. Homogeneous =−+ =+−− =−+ 086 0423 0453 321 321 321 xxx xxx xxx Nonhomogeneous −=−+ −=+−− =−+ 486 1423 7453 321 321 321 xxx xxx xxx 1 0 -4 3 0 0 1 0 0 0 0 0 0 1 0 -4 3 -1 0 1 0 2 0 0 0 0 = = −= freex x xx 3 2 31 0 3 4 = = −−= freex x xx 3 2 31 2 1 3 4 − = 1 0 3/4 3 3 2 1 x x x x − + − = 1 0 3/4 0 2 1 3 3 2 1 x x x x
- 22. Example 3: Describe all solutions for −=−+ −=+−− =−+ 486 1423 7453 321 321 321 xxx xxx xxx Solutions of Nonhomogeneous Systems i.e. Describe all solutions of whereAx=b − −− − = 816 423 453 A and b = 7 −1 −4 Geometrically, what does the solution set represent?